Assortative Mixing in Close-Packed Spatial Networks

Abstract

BackgroundIn recent years, there is aroused interest in expressing complex systems as networks of interacting nodes. Using descriptors from graph theory, it has been possible to classify many diverse systems derived from social and physical sciences alike. In particular, folded proteins as examples of self-assembled complex molecules have also been investigated intensely using these tools. However, we need to develop additional measures to classify different systems, in order to dissect the underlying hierarchy.Methodology and Principal FindingsIn this study, a general analytical relation for the dependence of nearest neighbor degree correlations on degree is derived. Dependence of local clustering on degree is shown to be the sole determining factor of assortative versus disassortative mixing in networks. The characteristics of networks constructed from spatial atomic/molecular systems exemplified by self-organized residue networks built from folded protein structures and block copolymers, atomic clusters and well-compressed polymeric melts are studied. Distributions of statistical properties of the networks are presented. For these densely-packed systems, assortative mixing in the network construction is found to apply, and conditions are derived for a simple linear dependence.ConclusionsOur analyses (i) reveal patterns that are common to close-packed clusters of atoms/molecules, (ii) identify the type of surface effects prominent in different close-packed systems, and (iii) associate fingerprints that may be used to classify networks with varying types of correlations.

In silico mutational studies of Hsp70 disclose sites with distinct functional attributes

“…Given any three‐dimensional structure of a self‐assembled molecular system, a network might be constructed according to the spatial arrangement of the selected nodes . For residue networks constructed from protein structures, an amino acid is considered as a point located on its C β atom ( C α atom in the case of glycine).…”

Local motifs in proteins combine to generate global functional moves

“…RNs are assortatively correlated networks in that k nn is an increasing function of k ( Figure 2A) [23,24]. Not only is this a manifestation of the hierarchical three-dimensional structure of proteins, but also are the two related by the third moment of A, the clustering coefficient C. C i is defined as the ratio of the number of actual to possible interconnections between neighbors of i, i.e.…”

“…one-third and increases up to one-half on the surface [2]. For a Poisson distributed network such as proteins, the relationship between the three moments of A was derived as in [24]…”

Combinatorial study of degree assortativity in networks

“…The understanding of the structure of these complex networks is vital for comprehending the evolutionary, functional, and dynamical processes taking place in these systems [4][5][6]. A major role in many of these processes, such as epidemic spreading, synchronization, percolation, social organization, protein architecture, network robustness, among others [7][8][9][10][11][12][13][14], is played by the degree assortativity [7]. A network is assortative if high-degree nodes tend to attach to other highdegree nodes, while it is disassortative if high-degree nodes tend to attach to low-degree ones.…”

“…For instance, many social networks, mainly collaboration networks, have been found to be assortative [7,9]. Also, close-packed spatial networks, such as protein residue networks, atomic (molecular) systems, and micellar networks are assortative [13,14]. Transitivity (clustering coefficients) [15,16] and modularity [17] are frequently found in empirical correlations with the assortativity coefficient [7,8,[18][19][20].…”